Frequency-dependent mass, elastic and geometric stiffness matrices of an axially loaded Timoshenko-Ehrenfest beam with applications

Earlier research on the development of explicit algebraic expressions for the elements of the frequency-dependent mass, elastic and geometric stiffness matrices for free vibration analysis was carried out on Bernoulli-Euler, Timoshenko-Ehrenfest and axially loaded Bernoulli-Euler beams. Seeking solution for the correspondingly more difficult problem for an axially loaded Timoshenko-Ehrenfest beam seemed too difficult at the time when these earlier developments took place. Now, with the experience and knowledge gained, the difficulty is overcome in part by enhanced application of symbolic computing. Thus, the explicit algebraic expressions for the elements of the frequency-dependent mass, elastic and geometric stiffness matrices of an axially loaded Timoshenko-Ehrenfest beam are derived from first principles. The equivalency of these matrices when added altogether, with the dynamic stiffness matrix is ensured. The derived matrices are then applied using the Wittrick-Williams algorithm as a solution technique to investigate the free vibration characteristics of some illustrative examples. The results are discussed, and significant conclusions are drawn. The proposed method preserves the exactness of results in the same way as the dynamic stiffness method, but importantly, it opens the possibility of including damping in free vibration and response analysis when using exact methods such as the dynamic stiffness method.